Non-Calculator Square Roots Up to 1960, and the
Computer Age, if you wanted to find square roots at all accurately, you either
used a mind-numbing �guess-and-check� long multiplication approach � or an
ingenious method attributed to Newton in about 1670. You�re going to use this
last bit of Fraction Wizardry � Step 1
To find the square root of, say, 5, we make a first guess � let�s call it
a1 = 2. We know this is an under-estimate, but that�s no problem Step 2 Since a square of area 5 has the same area as a rectangle of area 5 (not the genius part !) , we can use a1 and b1 as �lower� and �upper� estimates for
We deduce the value of
b1 from the fact that a1
x b1 = 5, so
Step 3 Since
a1 is an under-estimate, and
b1 is an over-estimate, we take their average as our
new, improved estimate for
� So, The Problem :A
Your first task is to repeat this procedure ( using
a2 instead of a1
) to find an even better estimate of
- I think you will be surprised how
good it is after even just these 2 repetitions. B Then try the same method, from scratch, to find a good estimate of the value of
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